Spectral Theory in Hilbert Space

More documents

Recommendations

Info

126 B. STIELTJES INTEGRALS We obtain n |s| ≤ |f(ξk)||g(xk) − g(xk−1)| k=1 ≤ max|f| [a,b] n k=1 |g(xk) − g(xk−1)| ≤ max [a,b] |f|V b a (g) . Since this inequality holds for all Riemann-Stieltjes sums, it also holds for their limit, which is b f dg. a In some cases a Stieltjes integral reduces to an ordinary Lebesgue integral. Theorem B.8. Suppose f is continuous and g absolutely continuous on [a, b]. Then fg ′ ∈ L1 (a, b) and b a f dg = b a f(x)g′ (x) dx, where the second integral is a Lebesgue integral. The proof of Theorem B.8 is left as an exercise (Exercise B.8).
EXERCISES FOR APPENDIX B 127 Exercises for Appendix B Exercise B.1. Prove Proposition B.1. Exercise B.2. Prove the calculation rules (1)–(5). Exercise B.3. Prove Proposition B.2. Exercise B.4. Show that if f and g has a common point of discontinuity in [a, b], then f is not Riemann-Stieltjes integrable with respect to g over [a, b]. Exercise B.5. Show that if f is absolutely continuous on [a, b], then f is of bounded variation on [a, b], and V b a (f) = b a |f ′ |. Hint: First show V b a (f) ≥ b a |f ′ |. To show the other direction, write (B.1) on the form b a ϕf ′ for a stepfunction ϕ and use Hölder’s inequality. Exercise B.6. Show that the set of all functions of bounded variation on an interval [a, b] is made into a normed linear space by setting f = |f(a)| + V b a (f). Convergence in this norm is called convergence in variation. Show that convergence in variation implies uniform convergence, and that the normed space just introduced is complete (any Cauchy sequence of functions in the space converges in variation to a function of bounded variation). Exercise B.7. Show that a monotone function can have at most countably many discontinuities, all of them jump discontinuities. Also show that if a function of bounded variation is continuous to the left (right) at a point, then so are its positive and negative variation functions, and that only if the function jumps up (down) will the positive (negative) variation function have a jump. Hint: How many jumps of size > 1/j can there be? Exercise B.8. Prove Theorem B.8. Also show that if g is absolutely continuous on [a, b], then any Riemann integrable f is integrable with respect to g and the same formula holds. Hint: f(ξk)(g(xk) − g(xk−1) = ϕg ′ where ϕ is a step function converging to f. Exercise B.9. Suppose f, g are continuous and ρ of bounded variation in (a, b). Put σ(t) = t f(s) dρ(s) for some c ∈ (a, b). Show that c b a g(t) dσ(t) = b a g(t)f(t) dρ(t) . Hint: Integrate both sides by parts, first replacing (a, b) by an arbitrary compact subinterval.
Page 1 and 2:
Spectral Theory in Hilbert Space Le
Page 3:
Preface The aim of these notes is t
Page 6 and 7:
iv CONTENTS Exercises for Chapter 1
Page 8 and 9:
2 0. INTRODUCTION cos( √ λ t), i
Page 10 and 11:
4 0. INTRODUCTION • Can the equat
Page 12 and 13:
6 1. LINEAR SPACES of u so one may
Page 14 and 15:
8 1. LINEAR SPACES Exercises for Ch
Page 16 and 17:
10 2. SPACES WITH SCALAR PRODUCT on
Page 18 and 19:
12 2. SPACES WITH SCALAR PRODUCT Pr
Page 20 and 21:
14 2. SPACES WITH SCALAR PRODUCT Ex
Page 22 and 23:
16 3. HILBERT SPACE Given a normed
Page 24 and 25:
18 3. HILBERT SPACE Two subspaces M
Page 26 and 27:
20 3. HILBERT SPACE ˆvn = limj→
Page 29 and 30:
CHAPTER 4 Operators A bounded linea
Page 31 and 32:
4. OPERATORS 25 Proposition 4.2. A
Page 33 and 34:
4. OPERATORS 27 In the rest of this
Page 35 and 36:
4. OPERATORS 29 must both be 0. Hen
Page 37 and 38:
CHAPTER 5 Resolvents We now conside
Page 39:
EXERCISES FOR CHAPTER 5 33 Analytic
Page 42 and 43:
36 6. NEVANLINNA FUNCTIONS However,
Page 44 and 45:
38 6. NEVANLINNA FUNCTIONS Proof. B
Page 46 and 47:
40 7. THE SPECTRAL THEOREM function
Page 48 and 49:
42 7. THE SPECTRAL THEOREM as The r
Page 50 and 51:
44 7. THE SPECTRAL THEOREM as a clo
Page 52 and 53:
46 8. COMPACTNESS weakly convergent
Page 54 and 55:
48 8. COMPACTNESS only if g ∈ L 2
Page 57 and 58:
CHAPTER 9 Extension theory We will
Page 59 and 60:
2. SYMMETRIC RELATIONS 53 We will n
Page 61 and 62:
2. SYMMETRIC RELATIONS 55 then it i
Page 63:
EXERCISES FOR CHAPTER 9 57 Exercise
Page 66 and 67:
60 10. BOUNDARY CONDITIONS the same
Page 68 and 69:
62 10. BOUNDARY CONDITIONS locally
Page 70 and 71:
64 10. BOUNDARY CONDITIONS of a sym
Page 72 and 73:
66 10. BOUNDARY CONDITIONS uniforml
Page 74 and 75:
68 10. BOUNDARY CONDITIONS Hint: If
Page 76 and 77:
70 11. STURM-LIOUVILLE EQUATIONS fu
Page 78 and 79:
72 11. STURM-LIOUVILLE EQUATIONS ob
Page 80 and 81:
74 11. STURM-LIOUVILLE EQUATIONS Ta
Page 82 and 83: 76 11. STURM-LIOUVILLE EQUATIONS Pr
Page 84 and 85: 78 11. STURM-LIOUVILLE EQUATIONS No
Page 86 and 87: 80 11. STURM-LIOUVILLE EQUATIONS Th
Page 89 and 90: CHAPTER 12 Inverse spectral theory
Page 91 and 92: 1. ASYMPTOTICS OF THE m-FUNCTION 85
Page 93 and 94: 2. UNIQUENESS THEOREMS 87 hand, the
Page 95 and 96: 2. UNIQUENESS THEOREMS 89 the bound
Page 97 and 98: CHAPTER 13 First order systems We s
Page 99 and 100: u(c) = 0 is given by (13.2) u(x) =
Page 101 and 102: 13. FIRST ORDER SYSTEMS 95 and (u2,
Page 103 and 104: 13. FIRST ORDER SYSTEMS 97 a non-ze
Page 105: EXERCISES FOR CHAPTER 13 99 Exercis
Page 108 and 109: 102 14. EIGENFUNCTION EXPANSIONS We
Page 110 and 111: 104 14. EIGENFUNCTION EXPANSIONS Ex
Page 112 and 113: 106 15. SINGULAR PROBLEMS We obtain
Page 114 and 115: 108 15. SINGULAR PROBLEMS We will g
Page 116 and 117: 110 15. SINGULAR PROBLEMS Proof. Le
Page 118 and 119: 112 15. SINGULAR PROBLEMS in the pr
Page 120 and 121: 114 15. SINGULAR PROBLEMS Since vK
Page 123 and 124: APPENDIX A Functional analysis In t
Page 125: A. FUNCTIONAL ANALYSIS 119 other wo
Page 128 and 129: 122 B. STIELTJES INTEGRALS (4) C (5
Page 130 and 131: 124 B. STIELTJES INTEGRALS Definiti
Page 135 and 136: APPENDIX C Linear first order syste
Page 137: C. LINEAR FIRST ORDER SYSTEMS 131 s
show all

Spectral Theory in Hilbert Space

Create successful ePaper yourself

Delete template?

Save as template?